Download 1 Ratio and Proportional Relationships, Grades 6 – 7 The work of

Ratio and Proportional Relationships, Grades 6 – 7
The work of teaching Ratio and Proportional Relationships
Ratios and proportional relationships are used to describe how quantities are
related and how quantities vary together. Their study bridges multiplication and
division of the elementary grades with linear functions in the upper middle grades
and high school.
Teachers may introduce ratios with juice or paint mixtures and walking distances
and times, for example. When two juices or paints are mixed, there are certain
quantities that yield mixtures that have identical flavors or colors. When people
walk, there are distances and times that correspond with the same pace of walking.
Such situations motivate the concepts of ratio and rate.
Teachers teach students to represent ratios in tables, on double number lines, and
with tape diagrams and to use ratio language (e.g., “A for every B,” “A parts to B
parts”) in discussing ratios. With the aid of these representations, students learn to
reason about quantities that are in equivalent ratios and they solve ratio problems
flexibly, and in increasingly sophisticated and abbreviated ways.
A central idea in the study of ratio and proportional relationships is the notion of a
(unit) rate, which tells the amount of one quantity per 1 unit of another quantity.
Teachers emphasize this rate language and they guide students toward problemsolving methods that rely on unit rates. As students work with proportional
relationships, in which two quantities vary together, teachers help students see that
a constant of proportionality, c, in an equation of the form y=cx and the slope of the
graph are unit rates.
Especially important is that teachers guide students to understand the distinction
between additive relationships and proportional relationships, which are commonly
confused. Teachers must draw students’ attention to wording such as “for every,”
“for each,” and “per,” which indicate a proportional relationship. They may also
highlight that “for each” and “per” refer implicitly to 1 unit of a quantity. For
example, 3/2 meters per second means 3/2 meters for every 1 second.
Ratios and proportional relationships are widely applicable. Teachers will teach
students to use them in geometry, when working with scale drawings, and in
statistics, when using samples to make inferences about populations. They will also
teach multi-step problems in which ratios are used, as in situations of percent
increase or decrease.
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Key understandings to support this work
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Know the meanings of ratio and rate and use ratio and rate language.
Explain how to use tables, double number lines, and tape diagrams to
represent and reason about ratios and equivalent ratios and to solve
problems.
Find unit rates in tables and on double number lines and use unit rates to
describe situations and solve problems.
Represent proportional relationships with tables, graphs, and equations and
explain how to relate the different representations.
Know how to determine when quantities described in a situation or with a
table or graph are or are not in a proportional relationship.
Recognize that a common error is for students to make an additive
comparison in cases where such a comparison does not apply.
Know ways of reasoning about and solving simple percent problems as well
as multi-step percent problems, such as those involving percent increase and
percent decrease.
Illustrative examples
Will a mixture of 1 cup blue paint with 3 cups yellow paint be the same color as a
mixture of 4 cups blue paint and 6 cups yellow paint? Why might sixth graders say
that these two paint mixtures will be the same color? Make two ratio tables, one for
each paint mixture, and use these tables to explain in several ways how the mixtures
compare.
To make “grapple juice,” we mix grape juice and apple juice in a ratio of 2 to 3.
Describe ways to interpret each of the fractions 2/3, 3/2, 2/5, and 3/5 in terms of
grapple juice (other than as ratios). [e.g., There is 2/3 of a cup of grape juice in the
mixture for every 1 cup of apple juice. There is 2/3 times as much grape juice in the
mixture as apple juice.] Show how these fractions occur in tables, equations, and
graphs relating quantities of juice.
If it takes 4 people 6 hours to mow a field, how long will it take 2 people to mow a
field of the same size? (Assume all the people work at the same steady rate.) Can we
solve this problem by setting up and solving a proportion, 4 people/6 hours = 2
people/H hours? Why or why not? How long would it take 3 people to mow a field of
the same size? Is that amount of time half way between the amounts of time for 4
people and 2 people or not?
Some pants are on sale for 20% off. The new, reduced price is $30. Why can’t you
find the original price by calculating 20% of $30 and then adding that to $30? How
can you reason correctly to find the original price?
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